Let \(\vartheta \)  be a congruence of an algebra \(\mathbf {A}\) .
We will often write \(a\equiv _{\vartheta } b\)  instead of \((a,b)\in \vartheta \) .
If \(\mathbf {B}\)  is a subalgebra of \(\mathbf {A}\) ,
we will say that \(\mathbf {B}\)  is saturated with respect
to \(\vartheta \), or \(\mathbf {B}\)  is a \(\vartheta \) -saturated subalgebra
of \(\mathbf {A}\) , if \(b\in B\)  and \(b\equiv _{\vartheta }a\)  imply
\(a\in B\)  for all \(a\in A\) .
In other words, \(\mathbf {B}\)  is \(\vartheta \) -saturated if and only if
its universe is a union of \(\vartheta \) -classes of \(\mathbf {A}\) .
For an arbitrary subalgebra \(\mathbf {B}\)  of \(\mathbf {A}\)  there exists a smallest
\(\vartheta \) -saturated subalgebra of \(\mathbf {A}\)  that contains \(\mathbf {B}\) , which we
denote by \(\mathbf {B}[\vartheta ]\) ; the universe of \(\mathbf {B}[\vartheta ]\) 
is \(B[\vartheta ]:=\bigcup _{b\in B}b/\vartheta \) .
Denoting the restrictions of \(\vartheta \)  to \(\mathbf {B}\)  and \(\mathbf {B}[\vartheta ]\)  by
\(\vartheta _{\mathbf {B}}\)  and \(\vartheta _{\mathbf {B}[\vartheta ]}\) , respectively,
we have by
the third isomorphism theorem ([1]})
that the map
\(\mathbf {B}/\vartheta _{\mathbf {B}}\rightarrow \mathbf {B}[\vartheta ]/\vartheta _{\mathbf {B}[\vartheta ]}\) ,
\(b/\vartheta _{\mathbf {B}}\mapsto b/\vartheta _{\mathbf {B}[\vartheta ]}(=b/\vartheta )\) 
is an isomorphism.
